Top

Mathematics in Computer Science

Published in:

01-12-2022

A New Type of Difference Dimension Polynomials

Author: Alexander Levin

Published in: Mathematics in Computer Science | Issue 4/2022

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

We introduce a new type of characteristic sets of difference polynomials using a generalization of the concept of effective order to the case of partial difference polynomials. Applying properties of these characteristic sets we prove the existence and find invariants of a bivariate dimension polynomial of a finitely generated difference field extension that describes the transcendence degrees of intermediate fields obtained by adjoining transforms of the generators whose orders lie between two given natural numbers. We also consider an application of the properties of introduced characteristic sets to the study of non-reflexive difference polynomial ideals.

previous article Using Semicontinuity for Standard Bases Computations

next article How Many Structure Constants do Exist in Riemannian Geometry?

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Blinkov, Y.A., Gerdt, V., Lyakhov, D.A., Michels, D.L.: On the consistency analysis of finite difference approximations. J. Math. Sci. 240(5), 665–677 (2019)CrossRefMATH

Cohn, R.M.: Difference Algebra. Interscience, New York (1965)MATH

Einstein, A.: The Meaning of Relativity. Appendix II (Generalization of Gravitation Theory), 4th edn., Princeton, pp. 133–165

Gerdt, V., Blinkov, Y., Mozzhilkin, V.: Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations. Symmetry, Integrability and Geometry: Methods and Applications, vol. 2, paper 051, 26 pp (2006)

Gerdt, V., Robertz, D.: Consistency of finite difference approximations for linear PDE systems and its algorithmic verification. In: Proceedings of ISSAC 2010. ACM Press, New York, pp. 53–59 (2010)

Gerdt, V., Robertz, D.: Computation of difference Gröbner bases. Comput. Sci. J. Moldova 20(2), 203–226 (2012)MathSciNetMATH

Gerdt, V., Robertz, D.: Algorithmic approach to strong consistency analysis of finite difference approximations to PDE systems. In: Proceedings of ISSAC 2019, ACM Press, New York, pp. 163–170 (2019)

Hrushovski, E.: The Elementary Theory of the Frobenius Automorphisms. arXiv:math/0406514v1, 2004, pp. 1–135. The updated version (2012). www.ma.huji.ac.il/ehud/FROB.pdf

Kolchin, E.R.: The notion of dimension in the theory of algebraic differential equations. Bull Am. Math. Soc. 70, 570–573 (1964)MathSciNetCrossRefMATH

10.

Kolchin, E.R.: Differential Algebra and Algebraic Groups. Acad Press, New York (1973)MATH

11.

Kondrateva, M.V., Levin, A.B., Mikhalev, A.V., Pankratev, E.V.: Differential and Difference Dimension Polynomials, Kluwer Acad. Publ. (1998)

12.

Levin, A.B.: Characteristic polynomials of filtered difference modules and of difference field extensions. Russ. Math. Surv. 33(3), 165–166 (1978)CrossRefMATH

13.

Levin, A.B.: Characteristic polynomials of inversive difference modules and some properties of inversive difference dimension. Russ. Math. Surv. 35(1), 217–218 (1980)CrossRef

14.

Levin, A.B.: Type and dimension of inversive difference vector spaces and difference algebras. VINITI (Moscow, Russia) 1606–82, 1–36 (1982)

15.

Levin, A.B.: Difference Algebra. Springer, New York (2008)CrossRefMATH

16.

Levin, A.: Bivariate dimension polynomials of non-reflexive prime difference-differential ideals. The case of one translation. In: Proceedings of ISSAC 2018. ACM Press, New York, pp. 255–262 (2018)

17.

Levin, A.: Bivariate Kolchin-type dimension polynomials of non-reflexive prime difference-differential ideals: the case of one translation. J. Symb. Comput. 102, 173–188 (2021)MathSciNetCrossRefMATH

18.

Levin, A.B., Mikhalev, A.V.: Type and dimension of finitely generated G-algebras. Contemp. Math. 184, 275–280 (1995)MathSciNetCrossRefMATH

19.

Mikhalev, A.V., Pankratev, E.V.: Differential dimension polynomial of a system of differential equations. In: Algebra (collection of papers), Moscow State Univ. Press, pp. 57–67 (1980)

20.

Wibmer, M.: Algebraic Difference Equations. Lecture Notes, University of Pennsylvania. http://www.mmrc.iss.ac.cn/mm2015/notes/wibmer1.pdf

Title: A New Type of Difference Dimension Polynomials
Author: Alexander Levin
Publication date: 01-12-2022
Publisher: Springer International Publishing
Published in: Mathematics in Computer Science / Issue 4/2022
Print ISSN: 1661-8270
Electronic ISSN: 1661-8289
DOI: https://doi.org/10.1007/s11786-022-00540-9

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 4/2022

30 Years with Vladimir Gerdt

Vladimir Gerdt: A Computer Algebra Enthusiast

How Many Structure Constants do Exist in Riemannian Geometry?

Personal Recollections of Prof. Vladimir P. Gerdt

Using Semicontinuity for Standard Bases Computations

Applications of Bar Code to Involutive Divisions and a “Greedy” Algorithm for Complete Sets

Premium Partner